Engineering Probability Class 17 Thurs 2018-03-22

Source

1   Exam 2, Thurs 3/29

  1. Bring 2 2-sided crib sheets.
  2. Bring a calculator if you wish. I'll try to set questions that don't require it. E.g., it's ok to write down an expression w/o evaluating it.
  3. The exam will include whatever normal distribution tables you need. It's legal if your calculator can do these, but I'll try to set questions where that doesn't help.

2   Chapter 5, Two Random Variables, ctd

  1. Read up to page 257.

  2. Review of some useful summations

    1. $$\sum_{k=0}^\infty a^k = \frac{1}{1-a}$$

      1. E.g. $$\sum_{k=0}^\infty 2^{-k} = 1 + 1/2 + 1/4 + \cdots = 2$$

      2. I'll prove it.

      3. E.g. for a geometric dist with $$f(k) = (1-p) p^k$$,

        $$\sum_{k=0}^\infty f(k) = 1 $$

        which is correct.

      4. That's Eqn 2.42b on page 64, and Example 3.15 on page 106, but with a different notation. Those use q where this uses p.

    2. $$\sum_{k=0}^\infty k a^k = \frac{a}{(1-a)^2}$$

      1. E.g. $$\sum_{k=0}^\infty 2^{-k} = 1/2 + 2/4 + 3/8 + 4/16 + \cdots = 2$$
      2. I'll prove it.
      3. For the geometric dist, the mean, $$\sum_{k=0}^\infty k f(k) = \frac{p}{1-p} $$
      4. That's Eqn 3.15 on page 106.

  3. Notation inconsistency.

    1. p in the geometric distribution here is called q in earlier chapters.
    2. The book often uses q=1-p. For these examples, q is the full number of blocks, q for quotient.

  4. Example 5.9 on page 242.

    1. The math is also relevant to filesystems where there is an underlying block size, like 4K. Some filesystems pack the partial last blocks of several files into one block.

  5. Example 5.12 on page 246.

  6. Section 5.3.1: Example 5.14 on page 247.

    This has mixed continuous - discrete random variables. The input signal X is 1 or -1. It is perturbed by noise N that is U[-2,2] to give the output Y.. What is P[X=1|Y<=0]?

  7. Example 5.15 on page 251. CDF of joint uniform r.v.

    $$f_{X,Y} = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp{\left(\frac{-{(x^2-2\rho x y + y^2)}}{{2(1-\rho^2)}}\right)}$$